Ultrafilter characterization of compact Hausdorff spaces ($\forall \mathcal{U}\exists !x:\mathcal{U}\to x$)

A topological space is both a compact space and a Hausdorff space exactly when every ultrafilter on its underlying set converges to exactly one point. Existence of all ultrafilter limits characterizes compactness. If a space is not Hausdorff, two distinct points have neighbourhood systems whose combined finite intersections are nonempty; the ultrafilter lemma extends these to an ultrafilter converging to both points. Conversely, disjoint neighbourhoods prevent such a common limit.